Summary of research activities

Math. Reviews for our Department

Per year:
2003 / 2002 / 2001 / 2000 / 1999 / 1998 / 1997 / 1996 / 1995 / 1994 / 1993 / 1992 / 1991 / 1990 / 1989 / 1988 / 1987 / 1986 / 1985 / 1984 / 1983 / 1982

Note: This is only approximate, especially for the period up to mid eighties.

See also the site:

Algebra - Number Theory - Algebraic Geometry

Groups of automorphisms of algebraic curves, torsion points of elliptic curves, elliptic curves with everywhere good reduction, inverse problem of Galois theory, representation of primes via quadraric forms.

Solution of equations in integers which define elliptic curves over Q with the method of forms of elliptic logarithms.

Vector bundles on Riemann surfaces, geometry of the moduli space who parametrizes semi-stable vector bundles, families of abelian varieties and theta functions, symmetric products of curves.

Combinatorics - Discrete Geometry

Enumerative and bijective combinatorics, combinatorics of partially ordered sets, matroid theory, hyperplane arrangements and face enumeration, combinatorics of root systems, Weyl groups and Coxeter arrangements.

Combinatorics of polytopes, monotone paths on polytopes, fiber polytopes, combinatorics and discrete geometry of zonotopes and arrangements of hyperplanes, triangulations and zonotopal tilings, oriented matroids.

Combinatorics and topology of simplicial and cell complexes, topology of partially ordered sets, combinatorics and topology of the refinement poset of polyhedral subdivisions.

Geometry - Topology

Dynamical Systems and Differential Topology and Geometry, as well as the interaction between them, i.e. chain recurrence and various rotation numbers for homeomorphisms, diffeomorphisms, flows and smooth volume preserving vector fields. Also, exploration of possible relations between dynamical invariants for classes of volume preserving vector fields on 3-manifolds.

The dependence of bending deformation from the geodesic lamination at which it is defined, study of Hyper-Kaehler geometry of space of quasi-fuchsian structures, study of deformations of conformal and projective structures in relation with the Teichmueller metric, problems of decidability in fields of meromorphic functions, use of computer to differential geometry.

Logic

The decidability problem for theories and existential theories of algebraic domains and analogues to Hilbert’s tenth problem especially for the following domains: polynomial rimgs, number rings, formal power series rings, rings of analytic functions (complex and/or p-adic), fields of rational functions, fields of meromorphic functions. The oustanding problem of the area is whether the existential theory of the field of rational numbers is decidable, that is, the analogue of Hilbert’s tenth problem for the rationals.

Foundations of Computer Science and abstract computability, effectiveness in the real numbers: model of computations are considered over real fields in a foundational setting, paralleling and supplementing the work of Blum-Smale.

Differential equations

Elliptic and parabolic problems: Blow-up study for solutions of nonlinear parabolic problems, calculus of variations for elliptic problems, reaction-diffusion problems, problems of regularity for free boundaries, problems of phase transition as singular limits of reaction-diffusion equations, qualitative theory for viscosity solutions of phase transition problems. Quasilinear non-uniform elliptic and parabolic problems: general solvability, blow-up of the gradient of the solution, convergence in the limit with repsect to small parameters.

Hyperbolic problems: Relaxation limits for the approximation of weak solutions for scalar conservation laws, geometric theory of shock waves for first order equations, viscosity solutions for Hamilton-Jacobi equations.

Application problems

Population genetics, phase transition problems, multiphase geometrical optics in underwater acoustics, direct and inverse problems in underwater acoustics, wave propagation in layered media.

Numerical Analysis

Finite elements or finite volume schemes for the equation of elastodynamics, conservation laws and wave propagation. Implicit-explicit schemes for quasilinear parabolic equations. Finite volume schemes for Hamilton-Jacobi equations.

Problems of numerical linear algebra such us solution of linear systems resulting from the discretization of partial differential equations, alternating Schwartz method. Computation of optimal parametres and field of convergence of classical methods of solution of linear systems. Domain decomposition methods, equivalence of recursive methods, semi-recursive methods of solutions of linear systems.

Scientific computations

Computations for underwater applications, for problems of high frequency wave propagation, for wave propagation problems in layered media.

Statistics

Construction and study of Generalized Shrin KagZ Estimators. We assume a generalized linear model, normal or spherically symmetric with covariance matrix partially or completely unknown. For the estimation of the location parameter, under squared error loss, the inadmissibility of the L.S.E., when the dimension of the parameter space is >3, leads to the search for better estimators in the class of the Generalized James-Stein estimators. These estimators are examined from several criteria such as: unbiased estimation of the risk, minimaxity and admissibility. Finally, these estimators are interpreted in a Bayesian Framework.

The problem of the prediction of a random parametric function. This problem is examined when the underlying distribution is continuous, more specifically, when it belongs to the expontential family. A natural identity that appeared first in Stein (1973), and has been widely exploited since, is discussed in relation with numbers of such a family. Mild conditions are also introduced that imply the nonexistence of a uniformly minimum mean squared error predictor (UMMSEP) for a random function.

Non-parametric estimation of functions and functionals from i.i.d. samples, censored data and sample paths of stochastic processes.

Analysis

Convex geometry: Classical and new problems on convex bodies are studied, concerning mainly: geometric inequalities, minima or maxima of several geometric quantities, asymptotic behaviour of geometric quantities (for large dimensions).

Functional Analysis: The following two topics are represented: Operator theory in Hilbert and Banach spaces, Geometry of Banach spaces.

Harmonic and Complex Analysis: The studied problems concern mainly: the position of roots of polynomials (the Sendov - Ilyeff conjecture) and convergence of Taylor series (limit points of Cesaro convergent series, convergence of subsequences of partial sums).

Packings and tilings: Problems of packings and tilings in Rⁿ are studied by harmonic analysis methods.